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Quantitative model for predicting formation and muscle compressibility in during contraction and stretch

Laura Causey, Stephen C. Cowin, and Sheldon Weinbaum1

Department of Biomedical Engineering, The City College of New York, New York, NY 10031

Contributed by Sheldon Weinbaum, April 20, 2012 (sent for review March 2, 2012)

Skeletal muscle is widely perceived as nearly incompressible despite perimysial compartments. It is the purpose of this paper to develop the fact that and lymphatic vessels within the endomysial and a theoretical framework and simplified anatomical model for perimysial spaces undergo significant changes in diameter and performing just such an analysis. length during stretch and contraction. These fluid shifts between Perhaps the most extensive experimental investigation of the fascicle and interstitial compartments have proved extremely diffi- volume changes mentioned above is attributable to Mazzoni et al. cult to measure. In this paper, we propose a theoretical framework (5) who studied entire cross-sections of the rat spinotrapezius based on a space-filling hexagonal fascicle array to provide pre- muscle, a muscle comparable to a portion of the trapezius muscle in humans. These investigators made detailed measurements of dictions of the displacement of blood and lymph into and out of the fi muscle’s and during stretch and contrac- the cross-sectional area of individual muscle bers, the thickness tion. We also use this model to quantify the distribution of blood and of the interstitial (perimysial) space between fascicles, and the initial lymphatic (IL) vessels within a fascicle and its perimysial space cross-sectional area of the ILs in planes transverse to the primary using data for the rat spinotrapezius muscle. On average, there are and venules in the perimysial space. This was done for both electrically excited muscular contractions and passive stretches 11 muscle fibers, 0.4 /venule pairs, and 0.2 IL vessels per of up to 20%. Their detailed measurements of fiber cross-sections fascicle. The model predicts that the blood volume in the endomysial within the fascicles showed small, unexplained departures from space increases 24% and decreases 22% for a 20% contraction and isovolumetry. Unfortunately, it was too difficult to make meas- stretch, respectively. However, these significant changes in blood ∼ urements of the TAs and CVs within the fascicles or estimate volume in the endomysium produce a change of only 2% in fascicle their changes in diameter during contraction and stretch. These cross-sectional area. In contrast, the entire muscle deviates from investigators did make detailed measurements of the width of isovolumetry by 7% and 6% for a 20% contraction and stretch, re- the perimysial space, which was observed to be of nearly uniform spectively, largely attributable to the significantly larger blood vol- thickness, except in regions where there was a primary arteriole/ ume changes that occur in the perimysial space. This suggests that venule pair and, in some cases, an intervening IL. Most in- arcade blood vessels in the perimysial space provide the primary terestingly, they observed that the size of the ILs increased during pumping action required for the filling and emptying of ILs during stretch and decreased during active contraction. This last obser- muscular contraction and stretch. vation supported their hypothesis promulgated by Skalak et al. (3) that the dilation and contraction of the primary arterioles and dimensions | spacing | resting | isovolumetric | deformation venules served as a pump to fill and empty the ILs, which are known to be devoid of active contractile elements. Subsequently, Mazzoni n the 17th century, Jan Swammerdam placed an isolated frog’s et al. (5) proposed that the increased diameters of the muscle fibers Ithigh muscle in an airtight syringe and measured the volume of during contraction also compressed adjacent IL vessels and the the muscle as it contracted and relaxed by observing the movement decreased fiber muscle diameter during stretch also opened the ILs. PHYSIOLOGY of a droplet of water in the end of the syringe, thus inventing the They were unable to establish the relative importance of these first plethysmograph. He observed that the volume of the muscle two mechanisms, an important objective of our theoretical model, did not change as it was excited. Thus, when a muscle is isolated so because the vascular volume of the fascicle was not examined. that blood neither enters nor exits the muscle, it functions iso- There are no detailed measurements of the change in the vas- volumetrically (1). Furthermore, because the bulk modulus of cular volume within a fascicle during contraction and stretch or of water is 2.3 GPa, its resistance to volumetric deformation is very the change in IL cross-section with the dilation and constriction of large relative to its resistance to shape distortion, and the primary arterioles and venules. The density of the ILs has also itself is often described accurately as “incompressible.” However, never been measured, and the fraction of primary arterioles and fi ENGINEERING the entire muscle and parts of the muscle may experience volu- venules that are accompanied by an IL has not been quanti ed. In metric changes attributable to the fact that both blood and lymph the absence of such measurements, we have constructed a quan- reside in highly flexible vessels in which they can both enter and titative anatomical model for a space-filling hexagonal fascicle exit with little resistance. These fluid conduits lie in both the array interspersed with a uniformly distributed perimysial space endomysial space between individual muscle fibers within fascicles that also contains local regions of enlargement to accommodate and in the perimysial space surrounding and separating the fas- the countercurrent arcade vessels. As a substitute for much of this cicles (Fig. 1A). The endomysial space contains , ter- missing information, we have combined the measurements on minal arterioles (TAs), and collecting venules (CVs). Initial individual microvessels, subject to tetanic excitation (6, 7) and lymphatic (IL) vessels are occasionally seen in the bed passive muscle length changes (8, 9), with the measurements of adjacent to muscle fibers; however, it is unclear if these vessels are the overall behavior of the fascicles and ILs in the study by involved in lymph formation and transport (2). The perimysial space contains arcade arterioles and venules (also called primary and secondary arterioles) and ILs, which lie in close proximity to Author contributions: L.C., S.C.C., and S.W. designed research, performed research, ana- the larger arterioles and venules (3, 4). Because the changes in lyzed data, and wrote the paper. blood and lymph volume between these compartments are very The authors declare no conflict of interest. small compared with the whole-muscle tissue volume, it has been 1To whom correspondence should be addressed. E-mail: [email protected]. fi dif cult to either measure these changes or quantitatively relate This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. the relative changes in volume between the endomysial and 1073/pnas.1206398109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1206398109 PNAS | June 5, 2012 | vol. 109 | no. 23 | 9185–9190 Downloaded by guest on September 26, 2021 resting state, these investigators provided the following frac- tional area ratios. The perimysial interstitium, blood vessels, and IL vessels, are 7.4%, 11.0%, and 0.4% of the total area of the muscle, respectively. To use this information, At = Af + Ai + Ab + Al is rewritten in the following form using the area ratios to provide a correlation between Af and Ai: A A A t A ¼ A þ A þ b A þ l A A i f i A i A i i i i [1] Ai 0:11 0:004 ¼ Af þ Ai þ Ai þ Ai: 0:074 0:074 0:074

The area of the hexagonal fascicles, Af, with sides of length 2b 2 is given by Af =6√3b ,andtheareaoftheinterstitial space, Fig. 1. Muscle architecture (A) and hexagonal model (B) of muscle fascicles Ai, between the hexagons is given by Ai = 12(bh), where h is the and tissue. half-height of the perimysial space, which was previously noted to be equal to ∼3 μm (5), and b is 1/2 of the length of each hexagonal side. Substituting 6√3b2, 12(bh), and ∼3 μm into Eq. Mazzoni et al. (5) to construct a more complete picture of muscle 1, one finds that there is a single unique value of b that satisfies behavior. This includes model predictions for the frequency of the all the foregoing conditions for a space-filling hexagonal fascicle arcade vessels and their adjacent ILs and quantitative estimates of array, namely, b = 38 μm. With this result, all the unknown in- the small changes in vascular volume that occur in the endomysial dividual area components of the muscle at rest can be de- 2 2 and perimysial spaces during muscular contraction and stretch, termined. Therefore, At = 18,482 μm , Af = 15,006 μm , Ai = 2 2 2 and their relation to IL volume changes. 1,368 μm , Ab = 2,034 μm , and Al =74μm . The number of capillaries for each fascicle, ncap, was found by Anatomical Background and Model multiplying the capillary density, ρ, of 1,263 capillaries per For simplicity, we consider muscle to be composed of four hier- square millimeter (10) by the total area, At, of each muscle archical structural levels. The myofibril is the lowest level con- hexagon. We find that each muscle fascicle is paired with ∼23 sidered. It contains the force-generating . A number capillaries. Each TA runs transverse to the muscle fascicle di- of myofibrils are bundled into a muscle fiber, which is encapsu- rection and gives rise to capillaries. The TAs are spaced in lated by a membrane, to form the second hierarchical roughly 1-mm increments along the entire length of the fascicle level or cell level. The muscle fibers are surrounded by connective (11), as shown in Fig. 2. Each TA is ∼1 mm in length and 7–13 tissue called endomysium and are grouped into fascicles to form μm in diameter (6). Using the mean diameter of 10 μm, the − the third hierarchical level. The fascicles are surrounded by the volume of a single TA is found to be ∼7.9 × 10 5 mm3. perimysium and bundled into a whole muscle, which is encap- CVs are located adjacent to the TAs and have a diameter of sulated by the , to form the highest hierarchy level or 9–18 μm. If we assume the mean diameter is 13.5 μm, a single − organ level. The hierarchical levels are shown in Fig. 1A. CV is found to occupy a volume of ∼1.4 × 10 4 mm3. There are An idealized space-filling model was developed for muscle ∼23 capillaries running parallel to the muscle fibers within each fascicles, in which it was assumed that the fascicles, in their fascicle. Each capillary is ∼6.5 μm in diameter. In Eq. 2, we have transverse plane, are uniformly arranged hexagons with sides 2b used the value of Af to find the volume percent that the blood and a uniform interstitial tissue space of thickness 2h between vessels occupy within each fascicle. The total volume percent of the fascicles (Fig. 1B). As noted in Fig. 1A, each muscle fascicle blood vessels is the sum of the volume percent of the TAs and consists of muscle fibers (not shown), the contracting component CVs occupying the fascicle (%VTC) plus the volume percent of of the muscle, capillaries, and endomysium. The IL vessels are the capillaries in the fascicle (%Vcap):

2 3 −3 2 6:5 × 10 mm 6 − − 23 × π × × 1mm7 6 7:9 × 10 5 þ 1:4 × 10 4 mm3 2 7 %Vb;f ¼ %VTC þ %Vcap ¼ 6 þ 7 × 100%: [2] 4 15; 006μm2 15; 006μm2 5 × 1mm × 1mm ð1; 000μm=1mmÞ2 ð1; 000μm=1mmÞ2

arranged adjacent to the arcading arteriole/venule pairs in the Evaluating Eq. 2, we found %VTC and %Vcap to be equal to perimysial space (3), which is represented in Fig. 1B by the wide 1.46% and 5.09%, respectively, meaning that the capillaries oc- black lines. An unknown fraction of arteriole/venule pairs is cupy more than threefold the volume of the TAs and that the accompanied by an adjacent IL. However, these locally enlarged total blood volume percent of a fascicle, %Vbf, is equal to 6.55%. regions occupy only a small length of the fascicle perimeter and are not shown in Fig. 1B for simplicity. Changes That Occur During a Stretch and Contraction. IL vessel. Mazzoni et al. (5) found that the lymphatic contraction volume Muscle Components at Rest. The basic space-filling hexagonal ratio (Vl,c/Vl) is 0.55 when the muscle shortens 20% during model, contained within the dashed line in Fig. 1B, is composed a contraction (Ll,c/Ll = 0.8) and that the lymphatic stretch vol- of a fascicle, perimysial tissue, blood vessels, and IL vessels. The ume ratio (Vl,s/Vl) is 1.57 when the muscle is stretched 20% (Ll,s/ area of each of these four domains (Af, Ai, Ab, and Al,re- Ll = 1.2). The IL cross-sectional area per fascicle after con- spectively) is summed to give the total area of the hexagon traction can be calculated using these values where Vl, Ll, and Al At: At = Af + Ai + Ab + Al. The individual areas Af, Ai, Ab,and represent the volume, length, and cross-sectional area of an IL Al were not reported by Mazzoni et al. (5). However, for the vessel in a resting muscle, respectively:

9186 | www.pnas.org/cgi/doi/10.1073/pnas.1206398109 Causey et al. Downloaded by guest on September 26, 2021 Table 1. Change in diameter of blood vessels on a tetanic contraction % increase in diameter Vessel Diameter, μm from control

† Arcading arteriole 43.3* 24.5 TA 7–13† 55† Capillary 6.5 Not detectable † † CV 9–18 38 † † Arcading venule 69.3 19.5 Fig. 2. TAs and capillaries within the muscle fascicle. CVs run parallel to TAs shown. The % increase in diameter from control for the arcading arterioles and venules was estimated by averaging the changes in diameter of the primary and secondary arcading arterioles and venules in the perimysial space. : Vl *Data from Engelson et al. (4). 0 55 † Al;ct ¼ ¼ 0:69 × Al: [3] Data from Marshall and Tandon (6). 0:80Ll Similarly, for a stretch, the final area would be Blood vessels in the endomysial space: 20% stretch. ∼131% of the IL area: As a muscle is stretched, the capillaries, TAs, and CVs experience a decrease in 1:57Vl size, albeit the TAs and CVs are the only vessels in the endomy- Al; st ¼ ¼ 1:31 × Al: [4] sium that actively decrease as a result of vasoconstriction because 1:20Ll there are no contractile elements in capillaries. According to the The change in cross-section of the IL vessel is related to the relationship between muscle length and capillary diameter change in cross-section of the adjacent muscle fascicles and ar- established by Poole et al. (9), the 23 capillaries within each fas- teriole and venule pairs in the perimysial space as will be de- cicle passively decrease in diameter from 6.5 μm to 5.8 μm when scribed in the next section. the muscle is stretched 20%. Assuming a capillary length of 1 mm Blood vessels. and that the capillaries maintain their length because of their Table 1 lists the control diameters of the blood ves- − sels within the muscle and the approximate percent increase they tortuosity, this is a volume change of 1.6 × 10 4 mm3. We assume experience on a tetanic contraction, as recorded by Marshall and that the TAs and CVs decrease in diameter by 16% using data Tandon (6). The arcading arterioles and venules are located in the obtained for primary and secondary arterioles (7) because no perimysial space of the muscle, whereas the TAs and CVs act as known study has evaluated the changes that occur during a mus- bridges from the endomysium to the perimysium. Unfortunately, cular stretch in the TAs and CVs themselves. Also, it is reasonable Marshall and Tandon (6) did not measure the length change of the to assume that their length stays the same because they are muscle as it contracted. We assumed that a tetanus contraction transverse to the muscle fibers. This means that the volume of the − − produces a change in diameter that is comparable to TAs changes from ∼7.9 × 10 5 mm3 to 5.6 × 10 5 mm3, a volume − the change that occurs when a muscle is contracted 20%. change of 2.3 × 10 5 mm3. The volume of the CVs changes from − − − Blood vessels in endomysial space: 20% contraction. The compress- 1.4 × 10 4 to 1.0 × 10 4 mm3, a volume change of 4.0 × 10 5 mm3. ibility of the muscle fascicle can be ascertained by finding the Therefore, the capillaries contribute approximately 2.5-fold the compressibility of the blood vessels within the endomysial space: volume change of the TA/CV pair within the fascicle during the capillaries, TAs, and CVs. The TAs increase their diameter by stretch. The percent change in fascicle volume is 1.5%, as given by

: × −3 mm 2 : × −3 mm 2 6 5 10 5 8 10 PHYSIOLOGY 6:3 × 10−5 mm3 þ 23 × π − π × 1 mm 2 2 %ΔVf; st ¼ × 100%: [6] 1mm 2 15; 006 μm2 × × 1mm 1; 000 μm

55% on a tetanic muscular contraction. Using the average di- However, the total percent change in blood volume in the

ameter provided in Table 1 and assuming that the TAs maintain fascicle is 22%, a percentage 15-fold larger than the fas- ENGINEERING their length, this means that the volume of the TAs changes from S2 − − cicle itself (see Eq. ). ∼7.9 × 10 5 mm3 to 1.9 × 10 4 mm3, a difference in TA volume of In contrast to the negligible change in diameter observed −4 3 1.1 × 10 mm . The CVs increase their diameter by 38%, during a contraction (6), the capillaries decrease in diameter −4 3 meaning that the volume of the CVs changes by 1.4 × 10 mm to during a stretch because of mechanical forces acting on them. × −4 3 × −4 3 2.7 10 mm , a difference of 1.3 10 mm . Therefore, the The TAs and CVs have a smaller change in diameter during TAs and CVs contribute to the volume change by roughly the same a stretch compared with a contraction. The smaller change in amount. A review of the literature reveals that no noticeable diameter of arterioles and venules makes the total percent change in diameter occurs in the capillaries as the muscle contracts change in blood volume slightly less for stretched muscle. (6). The percent change in fascicle volume is given by Blood vessels in the perimysial space: Contraction. As shown in Ta- − − ble 1, the mean luminal diameter of an arteriole in the perimysial 1:1 × 10 4 þ 1:3 × 10 4 mm3 μ μ %ΔVf;ct ¼ × 100% ¼ 1:6%:[5] space is 43.3 m and the diameter of the venule is 69.3 m. The 1 mm 2 average percent increase in diameter is 24.5% for arterioles and 15; 006 μm2 × × 1 mm 1; 000 μm 19.5% for venules in the perimysial space (6); therefore, the combined arteriole and venule change in cross-sectional area is In contrast, the total percent change in blood volume in 2,424 μm2, a 46% increase. the muscle fascicle is 24%, or 15-fold larger than the fascicle itself Blood vessels in the perimysial space: Stretch. For a 20% stretch, (see Eq. S1). arteriole diameter decreases about 27%, based on linear

Causey et al. PNAS | June 5, 2012 | vol. 109 | no. 23 | 9187 Downloaded by guest on September 26, 2021 interpolation of the data in the study by Welsh and Segal (7). Af × ð1:270 − 1:250Þ 0:46 × Ab We assume that the venules experience the same decrease in %Δ A ¼ þ × %: [8] t;ct A A 100 diameter as well because no study, to our knowledge, has t t evaluated the change in the size of the venules. Therefore, the Using the values for Af, At, and Ab, Eq. 8 can be rewritten as combinedarterioleandvenulechangeinareais2,450μm2, a 47% decrease. %Δ At;ct ¼½0:016 þ 0:051 × 100%: [9] Results Thus, the blood vessels in the endomysial space contribute to Muscle Components at Rest. The mean vessel and muscle fiber 1.6% of the change in the total muscle area, and the blood cross-sectional areas are provided in Table 2. These values were vessels in the perimysial space contribute slightly more than used to determine the number of muscle fibers as well as blood threefold that amount. and IL vessels per fascicle. The number of muscle fibers per From Eq. 9, the total area of the muscle (hexagon) increases fascicle, nfibers, was estimated by dividing the area of a fascicle by ∼7% attributable to the expansion of the blood vessels in both the mean area of a fiber. Because each muscle fiber occupies the perimysium and the endomysium during a 20% contraction. 2 1,333 μm , according to figure 5 of ref. 5, we divide Af by 1,333 These calculations provide evidence that the deviation from μm2 to find that there are ∼11 muscle fibers per fascicle, as- isovolumetry of muscle may be explained by the dilation of the suming that the fascicle is composed primarily of muscle fibers blood vessels. The percentage change in the area of an IL during (small blood volume). Because there are ∼23 capillaries per contraction and stretch can be estimated from the linear relation fascicle, we conclude that each muscle fiber is paired with ∼2 shown in figure 8 of ref. 5. For a 20% contraction or stretch, the capillaries, a value comparable to the capillary/fiber ratio of 1.3 fractional area of the IL decreases or increases 31% on average. measured in the spinotrapezius muscle (12). Thus, the IL area per fascicle, which is 74 μm2 in the resting 2 2 2 To establish the number of IL vessels per fascicle, nlymph,we state, decreases or increases by 23 μm to 51 μm and 97 μm , divided the area of the ILs per hexagon, Al, by the mean area of respectively. The change in area of the blood vessels is nearly a single lymphatic vessel. Because the area of the ILs per fascicle 1,000 μm2, and the difference in area of the fascicles is over is 74 μm2 and the mean area of an IL vessel is 402 μm2 (5), there 3,000 μm2. Although the changes in cross-sectional area of the is ∼0.2 IL per fascicle. This calculation was performed for the fascicles and blood vessels dwarf that of the ILs, the percentage arteriole/venule pair as well. Thus, dividing the mean area of the changes are similar, with the blood vessel percentage increase arteriole/venule pair, 5,243 μm2, by the area of the arteriole/ve- being ∼1.5-fold the IL vessel decrease. nule pair per fascicle, 2,034 μm2,wefind that the number of such (ii) Stretch. A similar calculation was performed for a muscle pairs per fascicle is ∼0.4 pair per fascicle. Thus, ∼1/2 of arteriole/ stretched to 20% of its original length. The percent change in venule pairs have an IL vessel adjacent to them. For every 10 fascicle volume (%ΔVf,st) attributable to the compression of its fascicles, one finds that there are four arteriole/venule pairs, two blood vessels was calculated to be 1.5% (Eq. 6). The cross-sec- lymphatic vessels, 230 capillaries, and 110 muscle fibers. tional area of the fascicle after it is stretched 20% is

Changes That Occur During Stretch and Contraction. (i) Contraction. ð1 − 0:015ÞVf Af;st ¼ ¼ 0:821Af ; [10] After the muscle is contracted 20%, the percent change in fas- 1:20Lf cicle volume attributable to the expansion of its blood vessels (%ΔVf,ct) was calculated to be 1.6%. The cross-sectional area of where Vf, Lf, and Af represent the volume, length, and cross- the fascicle after it is contracted 20% is sectional area of a fascicle in a resting muscle, respectively. If the fascicle were incompressible, the area of the fascicle would be ð1 þ 0:016ÞVf ∼0.833 Ao, a difference of only ∼1%. The combined difference in Af;ct ¼ ¼ 1:270 Af ; [7] 0:8Lf muscle area of the fascicle and its surrounding perimysial space can be written as where Vf, Lf, and Af represent the volume, length, and cross- sectional area of a fascicle in a resting muscle, respectively. If the Af × ð0:833 − 0:821Þ 0:47Ab A %Δ At;st ¼ þ × 100%: [11] fascicle were incompressible, the area of the fascicle, f,ct, would At At be ∼1.250 Af, a difference of only ∼2%. To find how the blood vessels affect the cross-sectional area of the total muscle area Using the values for Af, At, and Ab, Eq. 11 reduces to during a contraction, we sum the effects that the blood vessels have in the endomysial space (the compressibility of the fascicle) %Δ At;st ¼½0:010 þ 0:052 × 100%: [12] and the effects that the blood vessels have in the perimysial space and divide the sum by the original total area of the muscle: Thus, the blood vessels in the endomysial space contribute to 1.0% of the change in the total muscle area, and the blood vessels in the perimysial space contribute approximately fivefold that amount. Combined, the total area of the muscle decreases ∼6% due to its Table 2. Mean vessel and muscle fiber cross-sectional areas at change in blood volume during a 20% muscular stretch. rest A summary of the changes in cross-sectional area that occur in Component Symbol Cross-sectional area, μm2 the fascicle, blood vessels in the perimysial space, and lymphatics for a stretched muscle is given in Table 3 for each hexagonal Ā 2 IL vessel lymph 402* fascicle unit. In contrast to the very small 23-μm increase in Ā † Arcade arteriole art 1,473 lymphatic area for each hexagonal fascicle unit, the change in Ā ‡ 2 Venule ven 3,770 area of the blood vessels is nearly 1,000 μm and the difference Ā 2 Capillary cap 20 in area of the fascicles is over 3,000 μm , results very similar to fi Ā Muscle ber fiber 1,333* a 20% contraction. fi *Data from Mazzoni et al. (5). Fig. 3 illustrates our ndings. As the arterioles, venules, and † Data from Engelson et al. (4). fascicles increase in cross-sectional area during a contraction, ‡Calculated assuming venule diameter is 1.6-fold the arteriole diameter, as they compress the IL vessels. The opposite occurs as the muscle estimated by Marshall and Tandon (6). stretches.

9188 | www.pnas.org/cgi/doi/10.1073/pnas.1206398109 Causey et al. Downloaded by guest on September 26, 2021 Table 3. Change in cross-sectional areas of components of is that for a typical 20% stretch, there is a 22% decrease in blood a hexagonal fascicle unit during a 20% contraction and stretch volume in the endomysial space, which results in only a 1.5% Cross-sectional Cross-sectional Cross-sectional area decrease in fascicle volume. The corresponding predictions for Component area at rest, μm2 area contracted, μm2 stretched, μm2 a 20% contraction are a 24% increase in blood volume, with only a 1.6% increase in fascicle volume. In marked contrast, our Af 15,006 18,457 12,320 model predicts that a 20% stretch of the arcade arterioles and Ab 2,034 2,970 1,078 venules in the perimysial space leads to a 5.2% decrease in the Al 74 51 97 area of the basic hexagonal unit cell containing the fascicle, more than fivefold that of the endomysial vessels. Similarly, for a 20% contraction, these arcade vessel pairs provide a 5.1% increase in Discussion area of the basic hexagonal unit cell, more than threefold that of the endomysial vessels within the fascicle. These predictions It has been generally assumed since the mid-18th century that provide a crucial comparison of the relative importance of the skeletal muscle is a nearly isovolumetric organ during de- endomysial and perimysial spaces in lymphatic pumping, sug- formation and the small changes in blood volume during con- gesting that the perimysial vessels provide the primary pumping traction and stretch of the muscle could be neglected. Whereas action for the ILs. there are a number of studies on the behavior of individual blood A key observation in the study Mazzoni et al. (5) is that the IL vessels during muscle stretch and contraction, the distribution of cross-sectional area is only 0.4% of the muscle hexagon con- blood between the endomysial and perimysial spaces has not < fi taining the fascicle and 1/25 the cross-sectional area of the been quanti ed and their relative importance in lymphatic blood vessels in the perimysial space. However, a change of pumping has not been assessed. Similarly, although the number ∼45% in area of these arteriole and venule pairs in the peri- density of capillaries in skeletal muscle has been measured (10), mysial space results in an area change of ∼30% in their associ- there have been no quantitative studies as to how the blood ated IL. If the muscle behaved as a truly isovolumetric organ vessels and lymphatics are distributed relative to individual fas- during deformation and the vessels were not subject to vasoac- cicles. This includes the number of arcade arterioles and venules tive responses, the blood vessels and the IL vessels would exhibit in the perimysial space per fascicle and the fraction of these arcade pairs associated with an adjacent lymphatic. The idealized similar behavior during stretch and contraction. However, as space-filling hexagonal model for a fascicle proposed herein shown by Skalak et al. (3) and Mazzoni et al. (5), the opposite provides quantitative estimates for these parameters for the behavior is observed. Substantial increases in IL area are ob- spinotrapezius muscle in its resting state. A striking feature of served during stretch, and, conversely, substantial decreases are the model is that there is only one value of the perimeter of the observed during contraction. In marked contrast, the arcade hexagon 12b, where b = 38 μm, which satisfies all the component arterioles and venules exhibited a diametrically opposite behav- area constraints measured by Mazzoni et al. (5). This allows one ior. It is the resolution of this basic paradox that lies at the to predict the average area of a fascicle (∼15,000 μm2), the area of IL pumping. The vasoactive responses responsible for this of its perimysial space (1,368 μm2), the number of the perimysial behavior are described next. blood vessel pairs per fascicle (0.4), and their associated lym- phatic area per fascicle (74 μm2) in the spinotrapezius muscle Effects of Stretch. Our model predicts that as a muscle is of a mature rat. The model also predicts that only half of the stretched, the blood vessels are compressed, causing a change in arteriole/venule arcade pairs have an accompanying IL and that the volume of the muscle, and therefore an apparent com- each muscle fiber within the fascicle is associated with two capil- pressibility of the tissue. As noted above, the fascicles decrease laries. The fact that five fascicles feed each lymphatic is strikingly about 1.5% in volume because of the loss of blood in the ∼ similar to the arrangement of tubules feeding the cortical col- capillaries, TAs, and CVs, with 70% of this loss occurring in lecting duct (CCD), where five closely spaced tubules feed each the capillaries. The loss of blood to these vessels is a result of the fl CCD at the corticomedullary junction in both the rat and rabbit blood being displaced from the muscle and the diminished ow into the muscle.

(13). The IL and the CCD play similar roles in that both are PHYSIOLOGY low-pressure drainage structures for a much larger organ. Skeletal muscle capillaries’ tortuosity is a function of sarco- The foregoing quantitative estimates provide the basis for mere length (14, 15). As the muscle shortens, the length of the predictions for the total blood volume of the endomysial space, capillaries remains constant but the capillaries take on a tortuous and the distribution volume between TAs, CVs, and capillaries. arrangement because they are restricted to the space between Capillary volume is ∼3.5-fold the combined volume of the TAs muscle fibers (16). As the muscle is extended, the capillaries are and CVs, and, together, they comprise 6.55% of the fascicle pulled into a straight configuration, and an additional extension volume. The more important prediction for lymphatic pumping will cause the vessels to stretch in the longitudinal direction (17). In the spinotrapezius muscle, capillary tortuosity decreases sys- tematically up to a length of 2.6 μm, a 20% increase in ENGINEERING length from the resting sarcomere length of 2.17 μm (9). Therefore, the capillaries straighten but do not stretch or change in length at the range of muscle lengths considered in this paper. The change in volume of the capillaries considered herein was dependent on the change in capillary cross-section. Diminished blood flow is influenced by the constriction of upstream blood vessels. As a hamster retractor muscle (a muscle similar in shape to the spinotrapezius muscle) is stretched, feed and arterioles progressively constrict, reducing their di- ameter up to 23% and 25%, respectively, leading to a reduced blood flow to the muscle (7). According to the latter study, the passive lengthening initiates the release of norepinephrine from the sympathetic nerves, constricting the vessels. The researchers Fig. 3. Conceptual drawing of muscle cross-section stretched (A) and con- concluded that the majority of the vascular resistance occurs as tracted (B). The arteriole and venule pairs in the perimysial space expand a result of the sympathetic response, whereas the compression and compress the adjacent IL vessels during a stretch and a contraction, and buckling of the microvessels within the muscle attributable respectively. to a mechanical response account for the remainder.

Causey et al. PNAS | June 5, 2012 | vol. 109 | no. 23 | 9189 Downloaded by guest on September 26, 2021 The IL vessels are located adjacent to and in parallel with vasodilation is amplified through upstream dilation of vessels muscle fascicles and are connected to them by the perimysial by conducted dilation in which a response travels upstream tissue (3). Mazzoni et al. (5) have suggested that when skeletal along the vessel wall. Flow-mediated dilation then occurs as the muscle is stretched, the muscle fiber area decreases, pulling the response of endothelial cells to shear stress acting via release of interstitium surrounding the muscle. However, this area change NO, prostacyclin, and endothelial-derived hyperpolarizing is isovolumetric, because the muscle fiberitselfisincom- factor (21). Because the feed arteries do not reside within the pressible, and the small change in fascicle blood volume is skeletal muscle itself, they are not exposed to the vasodilators dominated by the much larger local contraction of the arcade in the interstitium, and are therefore highly dependent on blood vessels in the perimysial space, which experience a 47% conducteddilation(21).Othermetabolites, such as lactate, H+, + decrease in area. This large local contraction pulls the IL vessel O2,K , and adenosine, produce vasodilation of skeletal muscle open, increasing its diameter and creating a suction pressure blood vessels as well. These metabolites may interact with and within the vessel. Because these IL vessels have a wall that only complement each other, thereby increasing local blood flow consists of a single endothelial layer, they are easily deformed. (22). The increased blood volume during a muscular contraction This action creates a pressure more negative than that of the causes the arcade arterioles in the perimysial space to increase surrounding tissue and allows fluidtoenterthelymphaticvessel in volume, resulting in a larger cross-sectional area, and the through unidirectional primary valves dispersed throughout the adjacent lymphatic vessels to collapse to accommodate the vessel wall (18). The effect has been characterized as that of decreasing space. The extension of these concepts to other a Pasteur pipette (19). muscle types is discussed in SI Text, Other Muscle Types. The constricted arterioles and collapsed capillaries enhance the effect of the decreased muscle diameter. The outcome of Concluding Remarks constricted arterioles has been observed in a resting spino- The blood vessels and lymphatics provide access and egress for trapezius muscle. Constricted arterioles have wide-open lym- the body’s fluids, creating changes in the total muscle that are phatic vessels adjacent to them, whereas the opposite is true sometimes interpreted as an apparent compressibility of the when the arterioles are dilated (3). Therefore, the constriction of muscle. We predict that the muscle departs from isovolumetry blood vessels in both the endomysial and perimysial spaces by 6–7% as it contracts and stretches and that the ILs and within stretched muscles has an effect on lymph formation, with perimysial arcade vessels behave in a diametrically opposite the blood vessels in the perimysium playing the dominant role. manner. The blood vessels in the perimysial space change the total muscle volume three- to fivefold more than the blood Effects of Contraction. Blood flow to the muscle is determined by perfusion pressure and vascular tone. The flow in muscle vessels in the endomysium, suggesting that these vessels pro- fi vide the primary mechanism for lymphatic pumping. The increases immediately after the rst (20), fi and after the first few contractions, resistance is decreased (21). space- lling hexagonal model proposed herein for the fascicles Feed arteries, arterioles, and venules dilate because of sym- provides a simple idealized anatomical model with which one pathetic and local responses when the skeletal muscle contracts may quantitatively analyze volume changes in endomysium (6). Muscle mechanoreceptors and chemoreceptors trigger an and perimysium. increased cardiac output and redirect the flow to the skeletal muscle, causing an increase in perfusion pressure. Flow is also ACKNOWLEDGMENTS. This work was supported by National Aeronautics and Space Administration Grant NNX10AN20H, National Institutes of augmented by decreased vascular tone. The vasodilation in the Health Grant 769220011, National Science Foundation Grant PHY-0848491, vascular bed is thought to commence with the release of sub- and the Professional Staff Congress-City University of New York Research stances from skeletal muscle, endothelial cells, and RBCs. The Award Program.

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